Subgroups — #13270092

Order 1792.

426 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89652	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#11728384	C₂	1	1	yes	maximal, characteristic
4	C₇×#1380	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725612	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725528	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725528	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
9	C₂×#6455	—	2	1	no	—
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₇×#107	C₂²	1	1	yes	characteristic
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×#6392	C₂²	1	1	yes	characteristic
14	C₂×#6392	C₂²	1	1	yes	characteristic
15	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#89759	—	2	1	no	—
17	C₂×#6382	—	2	1	no	—
18	#89652	—	2	2	no	—
19	#89652	C₄	1	4	yes	complemented
20	#89324	—	2	1	no	—
21	(C₂×D₇)⋊C₁₆	—	2	2	no	—
22	#12868	—	7	1	no	maximal, Sylow
23	C₂²×C₄×D₇	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₄	—	2	1	no	—
26	C₂×C₇×C₄⋊C₄	—	2	1	no	—
27	C₂×(C₂×D₇)⋊C₄	—	4	1	no	—
28	C₂×(C₂×D₇)⋊C₄	—	4	1	no	—
29	C₂×C₄⋊Dic₇	—	2	1	no	—
30	C₂×C₄⋊Dic₇	—	2	1	no	—
31	C₂×C₄×Dic₇	—	4	1	no	—
32	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₇	—	2	1	no	—
34	C₂×C₈×D₇	—	2	1	no	—
35	C₂×C₈×D₇	—	2	1	no	—
36	C₂×C₈×D₇	—	2	1	no	—
37	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₇	D₄	1	2	yes	—
39	#6455	—	2	4	no	—
40	#6455	—	4	2	no	—
41	C₂×C₇×C₁₆	—	2	1	no	—
42	C₂×C₇×C₁₆	—	2	1	no	—
43	C₂×C₇×C₁₆	—	2	1	no	—
44	C₇×#107	—	2	1	no	—
45	C₇×#107	—	2	1	no	—
46	C₇×#107	—	2	1	no	—
47	C₇×#107	C₂×C₄	1	2	yes	—
48	C₇⋊#95	—	4	1	no	—
49	#6392	—	2	1	no	—
50	#6392	—	2	1	no	—
51	#6392	—	2	1	no	—
52	#6392	C₂×C₄	1	2	yes	—
53	C₂×C₇⋊C₁₆	—	2	1	no	—
54	C₂×C₇⋊C₁₆	—	2	1	no	—
55	C₂×C₇⋊C₁₆	—	2	1	no	—
56	#6382	—	2	4	no	—
57	C₂×#427	—	7	1	no	—
58	C₂×C₂²⋊C₁₆	—	7	1	no	—
59	#1808	—	7	1	no	—
60	#1380	—	7	1	no	—
61	#1380	—	7	1	no	—
62	#1380	—	7	1	no	—
63	#1380	D₇	1	1	yes	characteristic, complemented
64	C₂³×D₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	—	4	1	no	—
66	C₂²×C₄×C₇	—	4	1	no	—
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₇	—	4	1	no	—
69	C₂²×Dic₇	—	4	1	no	—
70	C₂²×Dic₇	D₈	1	1	yes	characteristic
71	C₂×C₄×D₇	—	2	1	no	—
72	C₂×C₄×D₇	—	2	1	no	—
73	C₂×C₄×D₇	—	2	1	no	—
74	C₂×C₄×D₇	—	2	1	no	—
75	C₂×C₄×D₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×D₇	SD₁₆	1	2	yes	—
77	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₇×C₈	#41	1	1	yes	characteristic
80	C₂×C₇×C₈	#41	1	1	yes	characteristic
81	C₂×C₇×C₈	#41	1	1	yes	characteristic
82	C₂×C₇×C₈	#41	1	1	yes	characteristic
83	C₇×C₄⋊C₄	—	2	2	no	—
84	C₇×C₄⋊C₄	—	4	1	no	—
85	C₇×C₄⋊C₄	—	4	1	no	—
86	C₇×C₄⋊C₄	—	4	1	no	—
87	(C₂×D₇)⋊C₄	—	4	4	no	—
88	(C₂×D₇)⋊C₄	—	8	2	no	—
89	C₄⋊Dic₇	—	2	2	no	—
90	C₄⋊Dic₇	—	4	1	no	—
91	C₄⋊Dic₇	—	4	1	no	—
92	C₄⋊Dic₇	—	4	1	no	—
93	C₄×Dic₇	—	4	4	no	—
94	C₂×C₇⋊C₈	—	2	1	no	—
95	C₂×C₇⋊C₈	—	2	1	no	—
96	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
97	C₈×D₇	—	2	2	no	—
98	C₈×D₇	—	2	2	no	—
99	C₈×D₇	—	2	2	no	—
100	C₈×D₇	—	2	2	no	—
101	C₇×C₁₆	—	4	1	no	—
102	C₇⋊C₁₆	—	4	1	no	—
103	C₂³×C₈	—	7	1	no	—
104	C₂×#117	—	14	1	no	—
105	C₂²×C₁₆	—	7	1	no	—
106	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
107	#427	—	7	4	no	—
108	#427	—	14	2	no	—
109	C₂×#107	—	7	1	no	—
110	C₂×#107	—	7	1	no	—
111	C₂×#107	C₂×D₇	1	1	yes	characteristic
112	C₂×#107	C₂×D₇	1	1	yes	characteristic
113	C₂×#106	—	14	1	no	—
114	#380	—	14	1	no	—
115	C₂²⋊C₁₆	—	14	2	no	—
116	#340	—	14	1	no	—
117	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
118	C₂²×D₇	—	2	1	no	—
119	C₂²×D₇	—	2	1	no	—
120	C₂²×D₇	—	2	1	no	—
121	C₂²×D₇	—	2	1	no	—
122	C₂²×D₇	—	2	1	no	—
123	C₂²×D₇	—	2	1	no	—
124	C₂²×D₇	#102	1	2	yes	—
125	C₂×C₄×C₇	—	4	2	no	—
126	C₂×C₄×C₇	—	8	1	no	—
127	C₂×C₄×C₇	—	8	1	no	—
128	C₂×C₄×C₇	—	8	1	no	—
129	C₂×C₄×C₇	—	8	1	no	—
130	C₂×C₄×C₇	—	8	1	no	—
131	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	#123	1	1	yes	characteristic
133	C₂×C₄×C₇	#123	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇	#137	1	1	yes	characteristic
136	C₂×C₄×C₇	#137	1	1	yes	characteristic
137	C₂×Dic₇	—	2	1	no	—
138	C₂×Dic₇	—	2	1	no	—
139	C₂×Dic₇	—	4	2	no	—
140	C₂×Dic₇	—	8	1	no	—
141	C₂×Dic₇	—	8	1	no	—
142	C₂×Dic₇	—	8	1	no	—
143	C₂×Dic₇	—	8	1	no	—
144	C₂×Dic₇	—	8	1	no	—
145	C₂×Dic₇	#102	1	2	yes	—
146	C₄×D₇	—	2	2	no	—
147	C₄×D₇	—	2	2	no	—
148	C₄×D₇	—	2	2	no	—
149	C₄×D₇	—	2	2	no	—
150	C₇×C₈	#117	1	1	yes	characteristic
151	C₇×C₈	#117	1	1	yes	characteristic
152	C₇×C₈	#123	1	1	yes	characteristic
153	C₇×C₈	#123	1	1	yes	characteristic
154	C₇⋊C₈	—	2	2	no	—
155	C₂³×C₄	—	7	1	no	—
156	C₂²×C₈	—	7	1	no	—
157	C₂²×C₈	—	7	2	no	—
158	C₂²×C₈	—	7	2	no	—
159	C₂²×C₈	—	14	1	no	—
160	C₂²×C₈	—	14	1	no	—
161	C₂²×C₈	—	14	1	no	—
162	C₂²×C₈	—	14	1	no	—
163	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
164	#117	—	14	4	no	—
165	#117	—	28	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	14	1	no	—
169	C₂×C₄⋊C₄	—	14	1	no	—
170	C₂×C₂²⋊C₄	—	28	1	no	—
171	C₂×C₂²⋊C₄	—	28	1	no	—
172	C₂×C₄²	—	28	1	no	—
173	C₂×C₁₆	—	2	1	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	14	1	no	—
177	C₂×C₁₆	—	14	1	no	—
178	C₂×C₁₆	—	14	1	no	—
179	#107	—	2	1	no	—
180	#107	—	2	1	no	—
181	#107	—	2	1	no	—
182	#107	—	7	2	no	—
183	#107	—	14	1	no	—
184	#107	—	14	1	no	—
185	#107	—	14	1	no	—
186	#107	C₄×D₇	1	2	yes	—
187	#106	—	14	4	no	—
188	#95	—	28	1	no	—
189	C₂²×C₇	C₂×#102	1	1	yes	characteristic
190	C₂²×C₇	#480	1	1	yes	characteristic
191	C₂²×C₇	#480	1	1	yes	characteristic
192	C₂²×C₇	#508	1	1	yes	characteristic
193	C₂²×C₇	#508	1	1	yes	characteristic
194	C₂²×C₇	#509	1	1	yes	characteristic
195	C₂²×C₇	#510	1	1	yes	characteristic
196	C₂×D₇	—	2	2	no	—
197	C₂×D₇	—	2	2	no	—
198	C₂×D₇	—	2	2	no	—
199	C₂×D₇	—	4	1	no	—
200	C₂×D₇	—	4	1	no	—
201	C₂×D₇	—	4	1	no	—
202	C₂×D₇	—	4	1	no	—
203	C₄×C₇	—	8	2	no	—
204	C₄×C₇	—	16	1	no	—
205	C₄×C₇	#381	1	1	yes	characteristic
206	C₄×C₇	#419	1	1	yes	characteristic
207	C₄×C₇	#483	1	1	yes	characteristic
208	C₄×C₇	#483	1	1	yes	characteristic
209	Dic₇	—	2	2	no	—
210	Dic₇	—	8	2	no	—
211	Dic₇	—	16	1	no	—
212	C₂⁴	—	7	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	7	1	no	—
216	C₂²×C₄	—	7	2	no	—
217	C₂²×C₄	—	7	2	no	—
218	C₂²×C₄	—	14	1	no	—
219	C₂²×C₄	—	14	1	no	—
220	C₂²×C₄	—	14	1	no	—
221	C₂²×C₄	—	14	1	no	—
222	C₂²×C₄	—	28	1	no	—
223	C₂²×C₄	—	28	1	no	—
224	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
225	C₂×C₈	—	7	2	no	—
226	C₂×C₈	—	14	1	no	—
227	C₂×C₈	—	14	1	no	—
228	C₂×C₈	—	14	2	no	—
229	C₂×C₈	—	14	2	no	—
230	C₂×C₈	—	14	2	no	—
231	C₂×C₈	—	14	2	no	—
232	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
233	C₂×C₈	#1157	1	1	yes	characteristic
234	C₂×C₈	#1157	1	1	yes	characteristic
235	C₂×C₈	D₄×D₇	1	1	yes	characteristic
236	C₂×C₈	#1159	1	1	yes	characteristic
237	C₂×C₈	#1161	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	14	2	no	—
243	C₄⋊C₄	—	28	1	no	—
244	C₄⋊C₄	—	28	1	no	—
245	C₄⋊C₄	—	28	1	no	—
246	C₂²⋊C₄	—	28	4	no	—
247	C₂²⋊C₄	—	56	2	no	—
248	C₄²	—	28	4	no	—
249	C₁₆	—	4	1	no	—
250	C₁₆	—	28	1	no	—
251	C₂×C₇	#1823	1	1	yes	characteristic
252	C₂×C₇	#2138	1	1	yes	characteristic
253	C₂×C₇	#2139	1	1	yes	characteristic
254	C₂×C₇	#2234	1	1	yes	characteristic
255	C₂×C₇	#2234	1	1	yes	characteristic
256	C₂×C₇	#2237	1	1	yes	characteristic
257	C₂×C₇	#2237	1	1	yes	characteristic
258	D₇	—	4	2	no	—
259	C₂³	—	7	2	no	—
260	C₂³	—	14	1	no	—
261	C₂³	—	14	1	no	—
262	C₂³	—	14	1	no	—
263	C₂³	—	14	1	no	—
264	C₂³	—	14	1	no	—
265	C₂³	—	14	1	no	—
266	C₂³	D₇×D₈	1	1	yes	characteristic
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	7	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	14	1	no	—
275	C₂×C₄	—	14	1	no	—
276	C₂×C₄	—	14	2	no	—
277	C₂×C₄	—	14	2	no	—
278	C₂×C₄	—	14	2	no	—
279	C₂×C₄	—	14	2	no	—
280	C₂×C₄	—	28	2	no	—
281	C₂×C₄	—	56	1	no	—
282	C₂×C₄	—	56	1	no	—
283	C₂×C₄	—	56	1	no	—
284	C₂×C₄	—	56	1	no	—
285	C₂×C₄	—	56	1	no	—
286	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#6446	1	1	yes	characteristic
288	C₂×C₄	#6457	1	1	yes	characteristic
289	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
290	C₂×C₄	#6478	1	1	yes	characteristic
291	C₂×C₄	#6481	1	1	yes	characteristic
292	C₈	—	14	2	no	—
293	C₈	#6435	1	1	yes	characteristic
294	C₈	#6446	1	1	yes	characteristic
295	C₈	#6455	1	1	yes	characteristic
296	C₈	#6457	1	1	yes	characteristic
297	C₇	#12868	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	14	2	no	—
299	C₂²	—	14	2	no	—
300	C₂²	—	14	2	no	—
301	C₂²	—	28	1	no	—
302	C₂²	—	28	1	no	—
303	C₂²	—	28	1	no	—
304	C₂²	—	28	1	no	—
305	C₂²	D₇×#102	1	1	yes	characteristic
306	C₂²	#89546	1	1	yes	characteristic
307	C₂²	#89654	1	1	yes	characteristic
308	C₂²	#89683	1	1	yes	characteristic
309	C₂²	#89687	1	1	yes	characteristic
310	C₂²	#89688	1	1	yes	characteristic
311	C₂²	#89691	1	1	yes	characteristic
312	C₄	—	8	2	no	—
313	C₄	—	14	2	no	—
314	C₄	—	16	1	no	—
315	C₄	—	56	2	no	—
316	C₄	—	112	1	no	—
317	C₄	#89436	1	1	yes	characteristic
318	C₄	#89574	1	1	yes	characteristic
319	C₄	#89577	1	1	yes	characteristic
320	C₄	#89634	1	1	yes	characteristic
321	C₂	—	28	2	no	—
322	C₂	#11726625	1	1	yes	characteristic
323	C₂	#11726936	1	1	yes	characteristic
324	C₂	#11726943	1	1	yes	characteristic
325	C₂	#11726968	1	1	yes	characteristic
326	C₂	#11726974	1	1	yes	characteristic
327	C₂	#11727077	1	1	yes	characteristic
328	C₂	#11727096	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇×D₈
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1380	C₂
socle	C₂³×C₇	C₂×D₈
prime radical	C₂³×D₇	D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1380	D₇
hypocenter	C₇	#12868
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₇×#1380	C₂	semidirect
5. #11725612	C₂	semidirect
6. #11725528	C₂	semidirect
7. #11725528	C₂	semidirect
19. #89652	C₄	semidirect
63. #1380	D₇	semidirect
297. C₇	#12868	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#89652	2
C₂×(C₂×D₇)⋊C₁₆	2
#11728384	2
C₇×#1380	2
#11725612	2
#11725528	2
#12868	7